Abstract
We conjecture that 2t - 1 specified sets of 2t - 1 elements are enough to define an SBIBD(4t - 1, 2t - 1, t - 1) when 4t - 1 is a prime or product of twin primes (corrigendum 6:62,1992). This means that in these cases 2t - 1 rows are enough to uniquely define the Hadamard matrix of order 4t.
We show that the 2t -1 specified sets can be used to first find the residual BIBD(2t,4t - 2, 2t - 1, t, t - 1) for 4t - 1 prime. This can then be uniquely used to complete the SBIBD for t = 1,2,3,5. This is remarkable as formerly only residual designs with λ = 1 or 2 have been completable to SBIBD.
We note that not any set of elements will do as Marshall Hall Jr found 13 sets from 19 which could not be completed to an (19,9,4).
Publication Details
Jennifer Seberry, On small defining sets for some SBIBD(4t-1, 2t-1, t-1), Bulletin of the Institute of Combinatorics and its Applications, 4, (1992), 58-62.